Question

A body of mass $M$ is attached to one end of a rod of mass $M$ and length $L$ . The entire system is rotated about an axis passing through the other end of the rod. The $M.I.$ of the system about die given axis of rotation is:

• A. $\frac{M{L}^2}{3}$
• B. $\frac{4}{3}M{L}^2$
• C. $2M{L}^2$
• D. $3M{L}^2$

Answer 1

The correct option is B $\frac{4}{3}M{L}^2$

Moment of inertia of rod about its centre of mass is determined by the general form of moment of inertia which is given by:

$I=\underset{0}{\overset{M}{\int }}{r}^{2}dm$

Now mass $m$ is termed into length element $dr$

$I=\underset{-L/2}{\overset{L/2}{\int }}{r}^2\times \frac{M}{L}dr$

$I=\frac{M{L}^2}{12}$

Now moment of inertia of rod about its one end is given by

$I_R=\frac{M{L}^2}{12}+M\times {\left(\frac{L}{2}\right)}^2$

$I_R=\frac{M{L}^2}{12}+\frac{M{L}^2}{4}$

$I_R=\frac{M{L}^2}{3}$

Moment of inertia of another mass M which is attached at the other end of the rod

$I=M{L}^2$

Total moment of inertia

$I_T=I_R+I$

$I_T=\frac{4M{L}^2}{3}$

Thus moment of inertia of a rod and mass about its one of the end is$\frac{4M{L}^2}{3}$.